Course syllabus - Probability

Objectives

Probability theory deals with models for random experiments, i.e. experiments where it is not possible to predict the outcome even if one has full control of the external circumstances. Many phenomena where random variation is involved can be described in terms of probabilities. Random models are used in finance e.g. for stock prices and option prices. The course is aimed at equipping the students with the skills required for probabilistic modelling of real world situations. The course provides an important theoretical base for further courses in the Analytical Finance program such as Methods of Statistical Inference, Stochastic Processes, Actuarial Mathematics and Econometrics.

Learning outcomes

At the end of the course the student is expected to be able to
- formulate and apply the probability definition, basis laws, the Law of Total Probability and Bayes´ rule.
- formulate definitions of and apply discrete random variables: probability distributions, expected value and variance; the binomial distribution, the geometric distribution, the hypergeometric distribution and the Poisson distribution; moments and moment-generating functions; Tschebysheff's Theorem.
- formulate definitions of and apply continuous random variables: density and distribution functions, expected value and variance; the uniform distribution, the normal distribution, the gamma distribution and the beta distribution; moments and moment-generating functions; Tschebysheff's Theorem.
- formulate definitions of and apply multivariate distributions: bi- and multivariate distributions, marginal and conditional distributions; independent random variables; the expected value of a function of random variables; the covariance of two random variables; the expected value and variance of linear functions of random variables; the multinomial distribution and conditional expectation.
- describe and apply different methods for finding the probability distribution for a function of random variables.
- formulate and apply the Central Limit Theorem.
- in both oral and written form explain reasoning and solutions to problems that are solved to achieve the learning objectives specified above.

Course content

Probability: Definition. Probability laws. The Law of total probability and Bayes´ rule. Discrete random variables: Probability function. Expected value, variance and standard deviation. Binomial, geometric, hypergeometric and Poisson distributions. Moment-generating functions. Continuous random variables: Probability density function and probability distribution function. Expected value. Uniform, normal, gamma and beta distributions. Tschebysheff's theorem.Multivariate probability distributions: Bivariate and multivariate probability distributions. Marginal and conditional distributions. IndependenCovariance and correlation. Multinomial and bivariate normal distributions. Conditional expectation. Functions of random variables. The Central Limit Theorem.

Tuition

Lectures combined with exercises. Continuous examination of problems and projects combined with written tests. Work with and presentation of written reports.

Specific requirements

Calculus of Several Variables 7.5hp or corresponding.

Examination

Seminar (SEM2), 1.5 credits, marks Pass (G)
Written examination (TEN1), 3 credits, marks Pass (G) or Pass with distinction (VG)
Written examination (TEN2), 3 credits, marks Pass (G) or Pass with distinction (VG)

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass, Fail